First principle structural determination of (B2O3)n (n = 1–6) clusters: Fromplanar to cageLifen Li and Longjiu Cheng Citation: J. Chem. Phys. 138, 094312 (2013); doi: 10.1063/1.4793707 View online: http://dx.doi.org/10.1063/1.4793707 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v138/i9 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

THE JOURNAL OF CHEMICAL PHYSICS 138, 094312 (2013)

First principle structural determination of (B2O3)n (n = 1–6) clusters:From planar to cage

Lifen Li (���) and Longjiu Cheng (���)a)

School of Chemistry and Chemical Engineering, Anhui University, Hefei, Anhui 230039,People’s Republic of China

(Received 18 November 2012; accepted 14 February 2013; published online 5 March 2013)

The structure of (B2O3)n clusters (n = 1–6) are investigated using the method combining the geneticalgorithm with density functional theory. Benchmark calculations indicate that TPSSh functionalis reliable in predicting the energetic sequences of different isomers of (B2O3)n cluster comparedto the high-level coupled cluster method. The global minimum (GM) structures of (B2O3)n clus-ters are planar up to n = 3, and cages at n = 4–6. A Td fullerene is found in the GM structure atn = 6. The stability of three-dimensional structures increases with the size of the cluster accord-ing to the analysis of the calculated atomization energy. Natural bonding analysis given by adap-tive natural density partitioning reveals delocalized π -bonding in the 4-membered and 6-memberedrings, and it is aromatic at the centers of cages and rings. © 2013 American Institute of Physics.[http://dx.doi.org/10.1063/1.4793707]

I. INTRODUCTION

Much attention has been attracted by boron and boron-based clusters because of their interesting physical and chem-ical properties. Boron is an element capable of formingstrong covalent bonds and possesses an astounding variety ofchemistry. Experimental and theoretical studies on the elec-tronic structure,1–4 chemical bonding,5–7 and spectroscopicproperties8–15 of bare boron clusters as well as doped boronclusters16–19 have been reported. By replacing pairs of neigh-boring C atoms with alternating B and N atoms, one can pro-duce graphite-like boron nitride sheets and nanotubes, quiteanalogous to nanotubes of carbon. A number of studies onthe molecular, electronic structures, the spectroscopic, andthe thermo-chemical properties of small boron oxide clus-ters have been reported. Anderson and co-workers20–25 ex-perimentally investigated the oxidation of boron clusters andthe subsequent reactions of the BnOm

+ cations with smallmolecules in the gas phase. Wang and co-workers14, 26 usedexperimental photoelectron spectroscopy coupled with quan-tum chemical calculations to probe the electronic structuresof small anions including B3O2

−, B4O2−, B4O3

−, B4O22−

and their neutral counterparts. Most of the theoretical stud-ies have focused on BnOm species whose relationship of nand m leaves stoichiometry,27–43 and diboron trioxide (B2O3)n

clusters received little attention. Diboron trioxide is not onlyan important material in the ceramic and glass technology,44

but also an interesting material from a solid-state-physicalpoint of view due to its optical characteristics.45 At normalpressure, B2O3 has a trigonal structure (B2O3-I) character-ized by a three-dimensional network of corner-linked BO3

triangles.46, 47 At high pressure, an orthorhombic modification(B2O3-�) is more stable. It consists of a framework of linkedBO4 tetrahedra.48

a)Author to whom correspondence should be addressed. Electronic mail:clj@ustc.edu. Tel./Fax: +86-551-5107342.

Many achievements have been made in the modeling ofoxides (e.g., Al2O3 and Fe2O3).49–52 Recently, Woodley53 re-ported the stable and metastable stoichiometric (X2O3)n clus-ters of boron, aluminum, gallium, indium, and thallium oxidedefined using classical interatomic potentials. The method iseffective to the ionic compound as Woodley reported. How-ever, the structure of boron oxide, as far as we know, isnot supposed similar with that of alumina since boron atomprefers to form covalent bond with oxygen atom.

Therefore, we adopted density functional theory (DFT)method combined with genetic algorithm (GA) to predict thestructure of (B2O3)n (n = 1–6) clusters. The results show thatthe global minimum structures of (B2O3)n vary with n fromplane (n = 1–3) to cage (n >3), which are indeed differentfrom those of (Al2O3)n. And a Td fullerene is found to be theglobal minimum of (B2O3)6.

II. COMPUTATIONAL DETAILS

A. Global optimization method

The low-energy isomers of (B2O3)n clusters were locatedby the combination of GA and DFT method, which has beensuccessfully applied in the structural prediction of a numberof systems.50, 54–59 All DFT calculations were accomplishedby the GAUSSIAN 09 package60 using the TPSSh functional.61

In global research of the potential energy surface, a small ba-sis set (3-21G) is used for saving time. After global optimiza-tion, the low-lying TPSSh/3-21G isomers are fully relaxed atTPSSh/6-311+G* level of theory.

B. Benchmark calculations

For boron clusters, different functionals of DFT meth-ods make great difference in predicting the energetic se-quences of different isomers.56, 62 To verify the reliability ofdifferent functionals in the prediction of (B2O3)n clusters, a

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094312-2 L. Li and L. Cheng J. Chem. Phys. 138, 094312 (2013)

I II III IV V VI

1

C2v, 0.00 D∞h, 0.06

C2h, 0.00 C3h, 0.50 Td, 2.25

C3h, 0.00 D3h, 0.47 C2, 1.05Cs, 0.92

C2v, 0.00 C1, 0.15 Th, 0.71 C1, 0.71

C3, 0.00 C2, 0.79 C1, 1.36

Td, 0.00 S6, 1.69 D2h, 1.84

2

3

4

5

6

C1, 0.57 Cs, 1.25 C1, 1.52

S4, 2.20

C2v, 2.53

FIG. 1. The low-energy isomers of (B2O3)n (n = 1–6) clusters. The struc-tures in column I correspond to the global minima; columns II–VI corre-spond to isomers that are higher in energy, and the symmetry and the relativeenergies in eV are labeled. O-red; B-pink.

benchmark calculation is carried out by comparing the rela-tive stability of the three isomers of (B2O3)2 (see structures inFigure 1) in different methods. Results of the benchmark cal-culation are given in Table I to compare different functionals(TPSSh, PBE0,63 M06,64 B3LYP,65 and BPW9166, 67) with thehigh-level coupled cluster method (CCSD(T)/aug-cc-pvtz).68

Note that TPSSh and PBE0 functionals are consistent withCCSD(T) method in relative stability of the three isomers.However, relative stability of the planar isomers (2II and

TABLE I. Comparison of single point energies for the three low-lying iso-mers of (B2O3)2 (see structures in Figure 1).a

Method 2I 2II 2III

CCSD(T)b − 550.4027577 0.55 2.25TPSSh − 551.326837 0.50 2.25PBE0 − 550.713299 0.57 2.16M06 − 551.090825 0.41 2.41B3LYP − 551.344355 0.30 2.94BPW91 − 551.261793 0.41 2.61

aEnergies for 2I are in atomic units, other energies are relative to this in eV. Results aresingle point energies with 6-311+G* basis sets for TPSSh/6-311+G* geometry.bThe basis set is aug-cc-pvtz for CCSD(T).

2III) is highly overestimated by M06, B3LYP, BPW91, andBP86 functionals (especially for B3LYP functional, wherethe relative energy of the planar isomer (2I) is undervaluedby even 0.69 eV (2.94–2.25) compared to the 3D isomer(2III)). TPSSh/6-311+G* method is in reasonable agreementwith CCSD(T)/aug-cc-pvtz method tending to underestimatethe planar structure 2II about 0.05 eV, which suggests thatTPSSh/6-311+G* method is reliable in predicting relativestability of different packings of (B2O3)n clusters.

III. RESULTS AND DISCUSSION

Combing the GA with DFT method, we obtained thelow-energy isomers for (B2O3)n (n = 1–6) clusters at theTPSSh/6-311+G* level. Figure 1 plots the low-energy iso-mers of (B2O3)n (n = 1–6) clusters. All the isomers are ver-ified to be true local minima by frequency check (except forthe 1II, which is a saddle point on the energy landscape). Theglobal minima (GMs) are planar (up to n = 3), and cage at n= 4–6, and the energy gap between the lowest-energy planarand cage isomers is relatively large except for n = 4. The elec-tronic state, symmetry, energy gap, and atomization energy ofstructures for each isomers of (B2O3)n (n = 1–6) clusters arelisted in Table II.

TABLE II. Electronic statea, symmetryb, energy gapc, atomization energyd

and NICS valuee of structures for (B2O3)n (n = 1–6) clusters.

N ESa Symmetryb �HLc Eat

d NICSe

1I 1A1 C2v 7.56 27.461II 1�g Dh 7.58 27.402I 1Ag C2h 7.74 28.81 −9.262II 1A′ C3h 7.55 28.562III 1A1 Td 3.86 27.68 −9.803I 1A′ C3h 7.44 29.58 −4.543II 1A1

′ D3h 5.71 29.43 −8.103III 1A′ Cs 7.09 29.28 −4.733IV 1A C2 7.67 29.234I 1A1 C2v 6.84 30.07 −6.464II 1Ag C1 6.79 30.03 −5.684III 1Ag Th 5.58 29.89 −5.774IV 1A C1 7.17 29.894V 1A S4 6.49 29.52 −0.475I 1A C3 7.19 30.48 −5.145II 1A C2 7.46 30.32 −2.105III 1A C1 7.09 30.20 −3.086I 1A1 Td 7.30 30.76 −2.346II 1A C1 7.63 30.66 −1.926III 1A′ Cs 7.47 30.55 −0.956IV 1A C1 7.35 30.50 −0.166V 1Ag S6 6.95 30.48 −1.606VI 1Ag D2h 5.77 30.45 −1.72

aElectronic state.bPoint group (symmetry).cEnergy gaps (eV) between the highest occupied molecular orbital and lowest unoccu-pied molecular orbital.dAverage atomization energy, where Eat = [E(B2O3)n − 2n*E(B) − 3n*E(O)]/n.eNucleus independent chemical shift values (ppm) in cluster centers at the TPSSh/6-311+G* level.

094312-3 L. Li and L. Cheng J. Chem. Phys. 138, 094312 (2013)

A. Geometry structures

(B2O3)1: the GM of (B2O3)1 (1I) is a V-shaped config-uration with the B-O bond-distances of 1.21 Å and 1.33 Åand the B-O-B bond angle of 139.3◦. Linear structure 1II is asaddle point, which is only 0.06 eV higher in energy than 1I.1III is the most stable structure of (Al2O3)1, which is 2.53 eVhigher in energy than 1I.69

(B2O3)2: 2I is planar with a 4-membered ring (BOBO)and two tails (-OBO) attached to the two B atoms of the rhom-bus. As well as a planar structure, 2II is constituted of a BO3

triangle and three boronyls (-BO). 2III in Td symmetry isthe most stable configuration of (Al2O3)2, which is 2.25 eVhigher in energy.50, 52, 69

(B2O3)3: In the lowest energy configuration of (B2O3)3

(3I), there are three tails (-OBO) attached to the planar hexag-onal B3O3 unit. 3II is 0.47 eV higher in energy than 3I, whichis the most stable cage structure of (Al2O3)3.50, 52, 69 3III lies0.92 eV higher in energy whereas 3IV lies 1.05 eV higher inenergy.

(B2O3)4: The GM (4I) is a cage configuration in C2v

symmetry, and stacks of two 6-membered (chairlike) ringsand two 8-menbered rings. 4I is consistent with the resultof Woodley.53 The second low-lying isomer (4II) is quasi-planar with two 6-membered rings and an 8-membered ring.4II is 0.15 eV higher in energy at TPSSh/6-311+G* while iseven more stable at B3LYP/6-311+G* by 0.22 eV. 4III(Th)

is 0.71 eV higher in energy. 4IV has a 4-membered ring, a 6-membered ring and a trigonal BO3 unit, which is the lowest-energy cage isomer of (Al2O3)4.50, 52, 69 4V is quasi-planarwith an 8-membered ring and 4 tails (-OBO) and is 2.20 eVhigher in energy.

(B2O3)5: The most stable structure of (B2O3)5 (5I) is achiral C3 cap. The quasi-planar structure 5II (C2) consistsof two 6-membered rings and one 4-membered ring and is0.79 eV higher in energy than 5I. 5III (C1) consists of two6-membered rings and one trigonal BO3 unit.

(B2O3)6: A fullerene cage 6I with tetrahedral symmetryis suggested as the most stable structure of (B2O3)6. It can beregarded as four 6-membered rings occupying the four ver-texes and another six B atoms capping the edges of the tetra-hedron, which consists of four 6-membered rings and four12-membered rings. 6II is a 3D structure with three chair-like hexagonal B3O3 units and a B2O2 planar rhombus. 6III isquasi-planar with three 6-membered rings and a trigonal BO3

unit. It is 1.25 eV higher in energy. 6IV (C1) is a quasi-planarstructure with three 6-membered rings. A S6 isomer (6V) isfound with 1.69 eV higher in energy. 6VI is a D2h cage with1.84 eV higher in energy.

B. Atomization energy

The calculated atomization energy (average interactionenergy per B2O3 formula unit in the cluster: Eat = [E(B2O3)n

− 2n*E(B) − 3n*E(O)]/n) versus cluster sizes (the number offormula units in the cluster) is plotted for the lowest-energy(quasi-)planar and cage isomers in Figure 2(a). It is clearlyseen that the atomization energy increases with cluster sizeincreasing. The atomization energy of (quasi-)planar struc-

1 2 3 4 5 6

0

1

2

27

28

29

30

31

Rel

ativ

e en

ergy

(eV

)

Number of B2O3 units, n

(b)

(Quasi-)planar Cage

Eat

(eV

)

(a)

FIG. 2. (a) Atomization energy per B2O3 unit (Eat) of B2O3 clusters with thelowest-energy structures of different families ((quasi-)planar structures andcages) as a function of the number of B2O3 units n, where Eat = [E(B2O3)n

− 2n*E(B) − 3n*E(O)]/n; (b) Relative energies between the lowest-energystructures of different families (planar and cage structures) as a function ofthe number of B2O3 units n.

tures is higher than that of cages at n ≤ 3, and the cages ex-ceed at n ≥4. This conclusion is more vividly shown in Fig-ure 2(b) which plots the relative energy between the lowest-energy structures of the two different groups.

Additionally, the energy gap (EHL) between the highestoccupied molecular orbital (HOMO) and lowest unoccupiedmolecular orbital (LUMO) is an important factor influencingthe structural stability. The EHL of all structures are large (3.86∼ 7.74 eV). 2III have relatively small EHL (3.86 eV), as aconsequence of the instability of its geometry (the tension istoo large because of the small bond angel).

C. Electronic structure

The low-lying isomers of (B2O3)n clusters are (quasi-)planar and cage, just like Cn and Bn clusters, which are re-sulted by delocalized electrons. Are there delocalized elec-trons in (B2O3)n clusters? Since delocalization always as-sociated with aromaticity, we focus on the aromaticity ofthe (B2O3)n clusters first. The nucleus independent chemi-cal shifts (NICS) value is a popular magnetic criterion ofaromaticity.70 Table II gives the NICS value at the centerof some isomers for (B2O3)n cluster. The results show thatmost of isomers are aromatic, and only a few isomers arenon-aromatic, but none of them is anti-aromatic. Besides, wenote that all GMs of (B2O3)n clusters are aromatic based onNICS values listed in Table II. To the (quasi-)planar struc-tures with one ring (2I, 3I, and 4V), the degree of aromatic-ity (NICS = −9.26, −4.54, and −0.47 ppm) reduce as thecluster size increase, as well as the prism-like structures (3II,4III, 6VI corresponding to the NICS value −8.10, −5.77,−4.55, and −1.72 ppm, respectively). The reason may bethat electron delocalization is harder at a larger ring. Sim-ilar to other fullerene-like structure,71 6I is aromatic too(NICS = −2.34 ppm).

According to the aromaticity of (B2O3)n cluster, we in-fer that there must be delocalized electrons among the clus-ters. In order to get insight into the delocalized orbital and

094312-4 L. Li and L. Cheng J. Chem. Phys. 138, 094312 (2013)

bonding style of (B2O3)n clusters, canonical molecular orbital(MO) analysis and Adaptive natural density partitioning (Ad-NDP) is adopted. AdNDP is a new theoretical tool developedby Zubarev and Boldyrev72 for analysis of chemical bond-ing and has been successfully applied recently to the analy-sis of chemical bonding in clusters71–73 and organic aromaticmolecules,73 as well as boron and gold clusters.54, 56, 72, 73 Inthe following we pick some representative structures to dis-cuss their electronic structures.

For n = 1, the V-shaped structure is more stable than thelinear structure. Why is the V-shaped structure more stable?Here, we first focus on the nature of the bonding of the lin-ear structure 1II (Figure 3(a)). The distance between the Batom and the adjacent terminal O atom in the linear struc-ture is 1.21 Å, and that between the B atom and the central Oatom is 1.31 Å. Note that B-O single bond in the B2O3 crys-tal is 1.37 Å, the B-O double bond in BO2 is 1.26 Å, and theB-O triple bond in BO molecule is 1.20 Å.46, 74 The chemicalbonding between B atom and the terminal O atom is a triplebond, and that between B atom and the central O atom is a sin-gle bond (the Lewis structure showed in Figure 3(a)). Theremust be delocalized electrons in the structure. Note that 1IIhas 24 valence electrons (3 × 2 + 6 × 3), with each boronatom contributing three valence electrons and each oxygenatom contributing six valence electrons. Eight electrons arelocalized along the four B-O σ -bonds. The canonical π -MOs(Figure 3(a)) show that there are 12 π -electrons delocalizedon the whole molecule in two vertical directions in 1II. All ofthe atoms of the linear structure are in sp hybridization. The

2 × 1c-2e LPs

ON=1.96 |e|

4 × 2c-2e σ

ON=1.99-2.00 |e|

4 × 2c-2e π

ON=2.00 |e|

2 × 3c-2e π

ON=1.96 |e|

(a)

(b)

MO=10,11 MO=14,15 MO=16,17

MO=11 MO=14

MO=10 MO=15 MO=16

MO=17

(d)

2 × 1c-2e LPs

ON=1.96 |e|

4 × 2c-2e σ

ON=1.99-2.00 |e|

4 × 2c-2e π

ON=2.00 |e|

2 × 3c-2e π

ON=1.96 |e|

(c)

B O B OO1.21 1.31

B B OOO1.21 1.30

139.3

FIG. 3. (a) Lewis representation and π -MOs of 1II; (b) molecular structureand π -MOs of 1I; (c) AdNDP localized bonding patterns of structure 1II; (d)AdNDP localized bonding patterns of structure 1I. B-blue, O-red.

remaining 4 electrons are two lone pairs (LPs) on the othersp hybrid orbital of the terminal oxygen. The bond lengths(1.21 Å and 1.33 Å) and the π -MOs of V-shaped structure(Figure 3(b)) are similar to the linear structure. Note that theangel of the V-shaped structure is 139.3◦, which indicates thatthe hybridization of the central oxygen atom is between spand sp2 hybridization. In order to confirm this speculation,we applied AdNDP analysis to the two isomers. According toAdNDP analysis, there are two LPs (occupied number (ON)= 1.96 |e|, which is close to the ideal limit of 2.00|e|), fourtwo-center two-electron (2c-2e) σ -bonds (ON = 1.99–2.00|e|), four 2c-2e π -bonds (ON = 2.00 |e|), and two 3c-2e σ -bonds (ON = 1.96 |e|) in 1II (Figure 3(d)). The results showthat the chemical bonding between B atom and the terminalO atom and that between B atom and the central O atom are,as the supposition to the Lewis structure, triple bonds and sin-gle bonds, respectively. 1I is a distortion of 1II, and the twohave similar bonding patterns (Figure 3(c)). The two electronsin x-y plane of central oxygen atom of 1I can be consideredas 3c-2e bond (ON = 1.96 |e|) or LP (ON = 1.82 |e|) whichprobes that the hybridization of the central O atom of the V-shaped structure is between sp and sp2. The reason why theV-shaped structure is favored more may be that the oxygenatom prefers more sp2 hybridization.

(B2O3)2 adopts a C2h ground state geometry with a 4-membered ring which is, generally speaking, unstable in co-valence compounds. Why is the 4-membered ring stable inthe GM of (B2O3)2 cluster? The nature of the bonding of 2Iis analyzed first. The B-O bond lengths in 2I are 1.21 Å forthe distance of the terminal B-O bonds and 1.33–1.40 Å forothers (Figure 4(a)). Compared with the triple bond length(1.20 Å) and the single bond length (1.35 Å), we suppose the

4 × 1c-2e LPs

ON = 1.95-1.96 |e|

10 × 2c-2e σ

ON = 1.95-2.00 |e|

4 × 2c-2e π

ON = 1.97-2.00 |e|

1 × 2c-2e π*

ON = 2.00 |e|

1 × 4c-2e π

ON = 1.95 |e|

4 × 3c-2e π

ON = 1.97-2.00 |e|

(a)

(b)

MO=20 MO=23

MO=30 MO=31

MO=26

MO=34

°°

°

BO

BO O

B

O

BO

1.21

1.33

1.35

1.40

1.41

133.6

O

100.5

FIG. 4. (a) Lewis representation and π -MOs of 2I; (b) AdNDP localizedbonding patterns of 2I. B-blue, O-red.

094312-5 L. Li and L. Cheng J. Chem. Phys. 138, 094312 (2013)

terminal B-O bonds are triple bonds and others are singlebonds (Figure 4(a)). Note that the distance between the triplebond is farther, the length of the single bond is longer. We con-firm that non-uniform distributed delocalized electrons mustbe existed in the structure. Figure 4(a) gives the π -MOs of 2I,which shows that there are six π orbitals in 2I (MO = 20, 23,26, 30, 31, 34). That is, there are 12 electrons delocalized inthe whole molecule. In order to get insight into the nature ofthe bonding in 2I, AdNDP analysis was adopted. The AdNDPanalysis reveals that 2I has four LPs (two on the sp hybridorbital of the terminal O atoms and two on the sp2 hybrid or-bital of the O atoms in the 4-membered ring), ten 2c-2e B-Oσ -bonds, four 2c-2e π -bonds on the B-O terminals in two ver-tical directions, four 3c-2e π -bonds, one 4c-2e π -bond on the4-membered ring and one 2c-2e π∗-bond (see Figure 4(b)).It is obviously that, as the Lewis structure shows, the termi-nal B-O bonds are triple bonds and others are single bonds.The five π bonds (ten π -electrons) in z-axis direction conju-gate with each other on the whole molecule. The two LPs inthe pz orbital of the oxygen atoms in sp2 hybridization of the4-membered ring combine into one π orbital and one π∗ or-bital. The π orbital conjugates with the two empty p orbitalof the two B atoms. As a result, the energy of the π orbitaldecreases by the conjugation and the 4-membered ring is sta-ble. The 4-membered ring is aromatic with the NICS value of−9.26 ppm.

3I is planar with a planar hexagonal B3O3 unit. The B-Obond lengths in 3I are 1.21 Å for the three terminal B-O bondsand 1.33 Å–1.38 Å for other B-O bonds. Therefore, the threeterminal B-O bonds are considered triple bonds and others aresingle bonds (Figure 5(a)). Again, the length is longer when

BO O

B BO

O

O O

BO

BO

B O

1.21

1.33

1.37

1.381.38

132.4120.0 °°°

6 × 1c-2e LPsON=1.90-1.96 |e|

15 × 2c-2e σON=1.98-2.00 |e|

6 × 2c-2e πON=1.90-2.00 |e|

6 × 3c-2e πON=1.95-2.00 |e|

| |

(b)

(a)

MO=31 MO=33 MO=34

MO=38 MO=39 MO=40

MO=49 MO=50 MO=51

| |

FIG. 5. (a) Lewis representation and π -MOs of 3I; (b) AdNDP localizedbonding patterns of 3I. B-blue, O-red.

the single bond is farther from the triple bond. We supposethere must be non-uniform distributed delocalized electronsin the molecule as the same as found in the GM of (B2O3)2.Thirty valence electrons are localized along the ten B-O σ -bonds and 18 electrons are delocalized in the whole moleculeaccording to the MO analysis (nine delocalized π orbitalsshown in Figure 5(a)). We applied the AdNDP analysis for thedetail information of chemical bonding in 3I. AdNDP anal-ysis shows that there are six LPs with three in the terminalO atoms and three in the O atoms of the 6-membered ring,fifteen 2c-2e σ -bonds, six 2c-2e π -bonds, six 3c-2e π -bonds,and three 3c-2e π -bonds (Figure 5(b)), which verifies the sup-position to the Lewis structure of the chemical bonding types.The eighteen π -electrons in z-axis direction delocalize on thewhole molecule which is similar to their delocalized π -MOsand explains the stability of the planar structure. Accordingto Huckel’s 4n + 2 rule for aromaticity, the 6-membered ringis aromatic with six delocalized π electrons which get furthersupport from the NICS value at the ring center (−4.54 ppm).

6I is a tetrahedron cage in Td symmetry. The structurecan be regarded as four 6-membered rings occupying the fourvertexes of the tetrahedron and another six O atoms at theedges linking the 6-membered ring. The B-O bond lengthsare 1.38 Å for the bonding in 6-membered rings and 1.37 Åfor the bridging bonding, which means that all the bonds in6I are single bonds. There are 144 valence electrons (3 × 12+ 6 × 18) in total, 72 electrons of which are localized alongthe 36 B-O σ -bonds. Figure 6(a) plots the 18 π -MOs, whichillustrate that there are 36 electrons are delocalized in thewhole molecule. The AdNDP bonding patterns are presentedin Figure 6(b). The AdNDP analysis shows that there are two

12 × 1c-2e LPs

ON=1.90 |e|

12 × 2c-2e σ

ON=1.99 |e|

18 × 3c-2e π

ON=1.99 |e|

6 × 1c-2e LPs

ON=1.86 |e|

24 × 2c-2e σ

ON=1.98 |e|

(b)

(a)

MO=64 MO=68,69,70 MO=75,76

MO=80,81,82 MO=91 MO=95,96

MO=77,78,79

MO=85,86,87

FIG. 6. (a) Structure and π -MOs of 6I; (b) AdNDP localized bonding pat-terns of 6I. B-blue, O-red.

094312-6 L. Li and L. Cheng J. Chem. Phys. 138, 094312 (2013)

kinds of LPs (one on the pz orbitals of the O atoms in 6-membered rings, and another on the sp2 hybrid orbitals of thebridging O atoms) with a total number of 18 (ON = 1.86–1.90 |e|), two kinds of σ -bonds (One belongs to 6-memberedrings, and another belongs to the edges of the tetrahedron)with a total number of 36 B-O σ -bonds (ON = 1.98–1.99 |e|)and 18 3c-2e B-O-B π -bonds (ON = 1.99 |e|). Verifying thesupposition to the Lewis structure, all of the bonds are sin-gle σ -bonds. The AdNDP also shows that all of the oxygenatoms are in sp2 hybridization. The 36 π -electrons are dividedinto two kinds, one occupies the p orbital of the oxygen insp2 hybridization in the 6-membered rings, and another occu-pies one of the sp2 hybrid orbitals of the oxygen in the edgesof the tetrahedron. The 36 π -electrons delocalized by theentire fullerene, forming the molecular orbitals presented inFigure 6(a). The AdNDP analysis suggests that 6I is afullerene, which is consistent with the NICS value of −2.34ppm in the center of the cage and −2.97 ppm in the center ofthe 6-membered ring.

D. Discussion

(B2O3)n is sesquioxide the same as (Al2O3)n, (Ga2O3)n,(In2O3)n. There are differences between (B2O3)n and(Al2O3)n, (Ga2O3)n, (In2O3)n clusters. From (B2O3)n to(Al2O3)n, (Ga2O3)n and (In2O3)n, the ionicity of bond be-tween two atoms increase, and the covalency decrease gradu-ally. Their cluster structures gradually transform from 2D to3D. The building-up principle of (B2O3)n is different fromthat of (Al2O3)n, (Ga2O3)n, and (In2O3)n clusters. At smallsize (n = 2 and 3), the GMs of (Al2O3)n, (Ga2O3)n, and(In2O3)n clusters favor cage and 3D while the GMs of (B2O3)n

clusters at that sizes are favor of planes. For n = 4–6, theGMs of (Al2O3)n, (Ga2O3)n, and (In2O3)n clusters are all 3D,however, the GMs of (B2O3)n clusters are cages. Polarizingnature of boron is very strong, and boron oxides can be re-garded as covalent compound. The covalent characteristic ofB-O bond results in the structural difference between boronoxides and other IIIA oxides clusters. The particularity of B-O bond makes (B2O3)n clusters have unique structures andproperties, which would be potentially applicable in the fu-ture. For the larger size of (B2O3)n clusters, cage configura-tion is supposed to be the GMs.

IV. CONCLUSION

In the present work, the geometric and electronic struc-tures and chemical bonding of a series of small boron ox-ide clusters (B2O3)n (n = 1–6) are investigated using themethod combining the GA with DFT method (TPSSh func-tional). The low-energy isomers are obtained. The GM struc-tures are planar at n = 1–3, and cage at n = 4–6. We focuson the electronic structure analysis of the GMs of (B2O3)1,(B2O3)2, (B2O3)3, (B2O3)6. At n = 1, the V-shaped configu-ration is more stable than the linear configuration because thecenter O atom prefers sp2 hybridization. At n = 2, the delo-calized 4c-2e bond explains the stability of the 4-memberedring. There are π electrons delocalized in the whole molecu-

lar of planar (B2O3)3 and Td cage (B2O3)6. The (B2O3)n clus-ters prefer 4-membered ring and 6-membered ring becauseof the stability lead by the delocalized π -bonds in the ringaccording to the natural bonding analysis given by AdNDPanalysis, which also shows that there are many nc-2e bondingpatterns in boron oxides. In summary, (B2O3)n clusters favor(quasi-)planar at small size and cage structure at large size. Itis due to the electron-deficient p orbital of B element and theelectron-rich activity of O element, which result in electronsdelocalized in the whole molecule.

ACKNOWLEDGMENTS

This work is supported by the National Natural ScienceFoundation of China (Grant Nos. 20903001, 21273008), bythe 211 Project and by the outstanding youth foundation ofAuhui University. The calculations are carried out on theHigh-Performance Computing Center of Anhui University.We acknowledge Professor Boldyrev for the AdNDP codes.

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